An analytical study of the self-induced inviscid dynamics of two-dimensional uniform vortices

The self-induced dynamics of a uniform planar vortex in an isochoric, inviscid fluid is analytically investigated, through a nonlinear singular integrodifferential equation which describes the time evolution of the Schwarz function of its boundary. In order to overcome the difficulties related to the nonlinearity, an approximate solution of that equation is proposed in terms of a hierarchy of linear singular integral equations. The approximate solution at the first order is here investigated, by applying it to a class of uniform vortices the kinematics of which is well known. A rich mathematical phenomenology has been found behind the approximate dynamics. In particular, the motion of certain branch points and the consequent changes in the algebraic structure of the Schwarz function appear to be its key features. The approximate dynamics is finally compared with the numerical simulation of the motion of a sample vortex, the shape of which is selected in such a way that important changes occur during its first eddy turn-over time. They are well captured by the present solution, which satisfactory agrees with the numerical simulation for times of the order of a quarter of the eddy turn-over one. The agreement becomes more and more qualitative at later times, up to the breakup of the analytical solution, that occurs roughly at three quarter of the eddy turn-over time.